Instrument for drilling dental root canals

ABSTRACT

An endodontic instrument comprising a tapered rod, the rod comprising an active portion, the active portion terminating at a distal tip end of the rod and defining a conical envelope, the conical envelope having a longitudinal axis that coincides with an axis of rotation of the instrument.

THE CROSS-REFERENCE TO RELATED APPLICATIONS

This is a Continuation Application of U.S. Ser. No. 15/710,869, filed on Sep. 21, 2017, which is a Divisional application of U.S. Ser. No. 14/651,677, filed on Jun. 12, 2015, which is a national phase application of International Application No. PCT/162013/001191, filed on Jun. 7, 2013, which claims the benefit of and priority to International Application PCT/162013/000108, filed on Jan. 30, 2013, which are herein incorporated by reference for all purposes.

TECHNICAL FIELD

The object of the present invention is an instrument for drilling dental root canals.

BACKGROUND OF THE INVENTION

The treatment of an infected dental root is carried out by extracting the pulp using special instruments, then by shaping the root canal using successive drilling procedures, traditionally carried out with instruments of varying size and conicity. The final operation consists of filling the root canal.

The shaping of the root canal consists initially of enlarging the canal in its crown and medial parts to permit, as a second step, easier treatment of the apical part of the canal by mechanical cleaning the infected tissues.

An instrument designed for drilling dental root canals generally has a tapered rod fitted into a handle to permit it to be driven manually or mechanically and comprising, on at least part of its length, helicoidal flutes with a constant or non-constant pitch, and having at least one cutting edge.

Used in continuous rotation, this type of instrument may have a tendency to screw itself into the canal. Apart from the screwing action, another problem occurring in the production of instruments for drilling root canals is that of the strength and flexibility of the instruments. Indeed, when the instrument is too flexible it may bend or break before the practitioner has been able to complete the operation and when the instrument is too rigid, it follows the curvature of the dental root canal only with difficulty.

Numerous instruments have been developed in response to these problems. Document EP 1 361 831 describes an instrument for drilling dental canals comprising a base, a cutting section and a guiding section, the cutting section being defined by an envelope of a cylindrical or conical shape, the longitudinal axis of which coincides with the axis of rotation of the instrument. The cutting section has clearance zones disposed set-back with respect to the envelope, alternating with drilling zones disposed on said envelope.

This alternating arrangement of zones on the envelope and set back from the envelope makes it possible to reduce the risk of screwing the instrument into the dental canal. Furthermore, the axis of the cutting section can be offset with respect to the axis of the envelope. This makes it possible to deepen the clearance zones and to make the evacuation of the debris during treatment more effective.

Document U.S. Pat. No. 7,955,078 describes an endodontic instrument for preparation of dental root canals comprising a body shaped to turn about an axis of rotation. The body has a centre of mass which is not located on the axis of rotation of the instrument, thus giving the impression that the instrument is undulating (“swaggering”) when it is rotated. An instrument such as this has greater flexibility and thus makes it possible to follow the complex curves of a dental root canal most effectively.

However, in these two documents the axis of the active part is offset with respect to the axis of rotation of the instrument over the whole length of said active part and in particular the axis of the point of the instrument is offset. This can generate a beating motion of the point within the canal. It thus becomes difficult to ensure optimum dimensioning of the canal during treatment, in particular in its apical portion. Moreover, a point with its axis offset also has the disadvantage of pushing the debris back towards the apical portion rather than evacuating it towards the top of the canal.

SUMMARY OF THE INVENTION

The aim of the present invention is to produce an instrument for drilling dental root canals which obviates the stated disadvantages. In particular, one aim of the present invention is thus to produce an instrument which is flexible while being strong, reliable and effective and which makes it possible at the same time to respect the initial path of the root canal to be treated and to ensure optimum dimensioning of the canal in its apical portion after treatment.

The object of the present invention is an instrument for drilling dental root canals.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawings schematically illustrate by way of example a plurality of embodiments of the instrument in accordance with the invention.

FIG. 1 illustrates a first embodiment of an instrument for drilling dental root canals in accordance with the invention.

FIG. 2 is a cross-sectional view at the line II-II of the instrument shown in FIG. 1.

FIG. 3 is a cross-sectional view at the line III-Ill of the instrument shown in FIG. 1.

FIG. 4 is a cross-sectional view at the line IV-IV of the instrument shown in FIG. 1.

FIG. 5 illustrates a second embodiment of an instrument for drilling dental root canals in accordance with the invention.

FIG. 6 is a cross-sectional view at the line VI-VI of the instrument shown in FIG. 5.

FIG. 7 is a cross-sectional view at the line VII-VII of the instrument shown in FIG. 5.

FIG. 8 illustrates a third embodiment of an instrument for drilling dental root canals in accordance with the invention.

FIG. 9 is a cross-sectional view at the line A-A of the instrument shown in FIG. 8.

FIG. 10 is a cross-sectional view at the line B-B of the instrument shown in FIG. 8.

FIG. 11 is a cross-sectional view at the line C-C of the instrument shown in FIG. 8.

FIG. 12 is a cross-sectional view at the line D-D of the instrument shown in FIG. 8.

FIG. 13 is a cross-sectional view at the line E-E of the instrument shown in FIG. 8.

FIG. 14 is a cross-sectional view at the line F-F of the instrument shown in FIG. 8.

DETAILED DESCRIPTION OF THE INVENTION

In a first embodiment shown in FIG. 1 the instrument in accordance with the invention comprises a rod 1 fitted at one of its ends 1 a in a handle 2 permitting either manual actuation of the instrument or preferably its engagement in a hand-held part providing mechanical driving of the said instrument. In particular, the instrument 1 is intended to be driven in rotation about its axis of rotation R.

The rod 1 has an active part 1 b extending to the other end 3—the point 3—of the rod 1. Said active part 1 b is preferably tapered and conical, narrowing to the point 3 of the rod 1. Alternatively, the active part 1 b or the whole rod 1 could be cylindrical rather than conical.

The active part 1 b has a polygonal cross-section (the sides of which are straight or curved) and comprises cutting edges. More particularly in this first embodiment, the active part 1 b has, over its whole length, a square cross-section 4 forming four cutting edges 5 a, 5 b, 5 c, 5 d defining between them four helicoidal flutes 6, one flute being the face defined between two successive cutting edges of the active part 1 b. The active part 1 b is defined by an envelope 7 which is substantially tapered and has its longitudinal axis coinciding with the axis of rotation R of the instrument.

The particular feature of the instrument in accordance with the invention resides in the fact that the active part 1 b has a first portion 1 c extending from the point 3 towards the rear of the active part 1 b and of which the centre of mass is located on the axis of rotation R of the instrument and a second portion 1 d extending from the end of the first portion 1 c to the rear of the active part 1 b and of which at least one cross-section has a centre of mass which is not located on the axis of rotation R of the instrument but is offset with respect to said axis R. In the first embodiment shown in FIGS. 1 to 4, any cross-section of the second portion 1 d of the active part 1 b of the instrument 1 has a centre of mass which is not located on the axis of rotation R but is offset with respect to said axis.

More precisely, and as shown in FIG. 2, in accordance with the invention any cross-section 4 a of the first portion 1 c has its centre of mass ma on the axis of rotation R of the instrument. Moreover, in this first embodiment the four edges 5 a, 5 b, 5 c, 5 d of such a cross-section 4 a are located on the envelope 7. Thus the first portion 1 c and in particular the point 3 are centred with respect to the axis of rotation R of the instrument or in other words the longitudinal axis of the first portion 1 c coincides with said axis of rotation R.

As shown in FIGS. 3 and 4, in the first embodiment, any cross-section 4 b of the second portion 1 d of the active part 1 b has its centre of mass mb offset with respect to the axis of rotation R of the instrument and preferably a single cutting edge 5 a of such a cross-section 4 b is located on the envelope 7, the other cutting edges 5 b, 5 c and 5 d being disposed inside said envelope 7. Thus in the first embodiment the whole of the second portion 1 d of the active part 1 b is off-centre with respect to the axis of rotation R of the instrument.

Thus an effective instrument is obtained because its point 3 is centred, does not generate any beating within the canal and has four active cutting edges.

Such an instrument in accordance with the invention can be obtained from a rod with a circular cross-section, preferably made from a nickel-titanium alloy, by providing therein helicoidal flutes by machining (milling), these flutes defining cutting edges such that the cross-section of the rod 1 is polygonal over the whole length of its active part 1 b. In order to achieve the particular geometry of the active part 1 b in accordance with the invention, the flutes 6 of the second portion 1 d of the active part 1 b are overcut with respect to the flutes 6 of the first portion 1 c in order to obtain at least one cross-section 4 b of said second portion 1 d of which at least one cutting edge is set back within the envelope 7 and of which the centre of mass mb is offset with respect to the axis of rotation R. Thus on the second portion 1 d of the active part 1 b material is removed from the instrument in accordance with the invention, which makes it more flexible on this second portion than a traditional instrument which would have, over its whole active part, a cross-section with its centre of mass centred on the axis of rotation and all its edges inscribed on the envelope. By virtue of the present invention, an instrument is obtained which is effective at its point 3 while being flexible.

Preferably, in the first embodiment, the second portion 1 d has a progressive offset with respect to the axis of rotation R in the direction of the rear of the instrument: i.e. a cross-section of the second portion 1 d close to the point has its centre of mass less offset proportionally to the surface of the cross-section with respect to said axis of rotation R than a cross-section of the second portion 1 d closer to the rear of the instrument. In terms of machining the instrument, this produces flutes 6 which are overcut proportionally more and more along the second portion 1 d of the active part 1 b with respect to the flutes of the first portion 1 c. Thus in this first embodiment the flexibility of the instrument is adjustable and in particular increases progressively towards the rear of the active part 1 b.

Preferably, the first portion 1 c of the active part extends over a length of 3 millimetres starting from the point 3 of the active part 1 b. In a still more preferred manner, said first portion 1 c has a length of 1 millimetre.

FIGS. 5 to 7 show a second embodiment of an instrument in accordance with the invention in which the active part 1 b has, over its whole length, a rectangular cross-section 4′ forming four cutting edges 5 a, 5 b, 5 c, 5 d defining four helicoidal flutes 6, one flute being the face defined between two successive cutting edges of the active part 1 b. As in the first embodiment, the active part 1 b is defined by an envelope 7 which is substantially tapered and has the axis of rotation R of the instrument as its longitudinal axis.

The active part 1 b also has a first portion 1 c extending from the point 3 towards the rear of the active part 1 b and having its centre of mass located on the axis of rotation R of the instrument and a second portion 1 d extending from the end of the first portion 1 c to the rear of the active part 1 b and of which at least one cross-section has a centre of mass offset with respect to the axis of rotation R of the instrument. Preferably, and as in the first embodiment, any cross-section of the second portion 1 d of the instrument in accordance with the second embodiment has a centre of mass offset with respect to the axis of rotation R.

In this second embodiment, and as shown in FIG. 6, any cross-section 4′a of the first portion 1 c is square and has its centre of mass m′a on the axis of rotation R of the instrument. Furthermore, the four edges 5 a, 5 b, 5 c, 5 d of such a cross-section 4′a are located on the envelope 7 of the instrument. As shown in FIG. 7, any cross-section 4′b of the second portion 1 d of the active part 1 b has its centre of mass m′b offset with respect to the axis of rotation R of the instrument. In contrast to the first embodiment, for each cross-section 4′b of the second portion 1 d of the active part 1 c of the instrument in accordance with the second embodiment, two cutting edges 5 a and 5 b are on the envelope 7, the two other cutting edges 5 c and 5 d being disposed inside said envelope 7.

The other considerations relating to the first embodiment remain valid for this second embodiment.

As shown in FIGS. 6 and 7, the cross-sections 4′a and 4′b of the first and second portions 1 c, 1 d are not necessarily symmetrical, the cross-section 4′a preferably being square, while the cross-section 4′b may be rectangular.

FIGS. 8 to 14 show a third embodiment of an instrument in accordance with the invention, in which the active part 1 b of the instrument has, over its whole length, a polygonal cross-section 40 of a parallelogram shape, forming four cutting edges 51, 52, 53, 54 defining four helicoidal flutes 6, one flute being the face defined between two successive cutting edges of the active part 1 b. As in the previous embodiments, the active part 1 b is defined by an envelope 7 which is substantially tapered and has the axis of rotation R of the instrument as its longitudinal axis.

In accordance with the invention, the active part 1 b has a first portion 1 c extending from the point 3 towards the rear of the active part 1 b and having its centre of mass located on the axis of rotation R of the instrument. In this third embodiment, and as shown in FIG. 9, any cross-section 40 a of the first portion 1 c is in the form of a parallelogram and has its centre of mass m₄₀a on the axis of rotation R of the instrument. Moreover, in this embodiment, two diagonally opposed cutting edges 52, 54 of such a cross-section 40 a are located on the envelope 7 of the instrument, while the other pair of diagonally opposed cutting edges 51, 53 is located set back within the envelope 7. In accordance with the invention, the active part 1 b also has a second portion 1 d extending from the end of the first portion 1 c to the rear of the active part 1 b, of which at least one cross-section has a centre of mass which is not located on the axis of rotation R of the instrument but which is offset with respect to said axis R. In the third embodiment of the invention, and as shown in FIGS. 8 and 10 to 14, the second portion 1 d has an alternating arrangement of first zones—off-centre zones 11—in which any cross-section 401 b has a centre of mass m₄₀₁b offset with respect to the axis of rotation R of the instrument and second zones—centred zones 12—in which any cross-section 402 b has a centre of mass m₄₀₂b located on the axis of rotation R of the instrument. The zone of the second portion 1 d directly adjacent to the first portion 1 c of the active part 1 b of the instrument is an off-centre zone 11 (see FIG. 8).

FIGS. 11 and 13 each show a cross-section 402 b of a centred zone 12 located along the second portion 1 d of the active part 1 b of the instrument, while FIGS. 10, 12 and 14 each show a cross-section 401 b of an off-centre zone 11 located along said second portion 1 d.

Preferably, and as shown in FIGS. 11 and 13, for each cross-section 402 b of a centred zone 12, two diagonally opposed cutting edges 52, 54 are located on the envelope 7 of the instrument, while the other pair of diagonally opposed cutting edges 51, 53 is located set back within the envelope 7.

In a similar, preferred manner, and as shown in FIGS. 10, 12 and 14, for each cross-section 401 b of an off-centre zone 11, a single cutting edge 52, 54 appertaining to the pair of diagonally opposed cutting edges 52, 54 located on the envelope along the first portion 1 c of the active part 1 b of the instrument is located on the envelope 7 of the instrument, while the other cutting edges are located set back within the envelope 7.

Thus, apart from the alternation between the centred and off-centre zones 12,11 on the second portion 1 d of the active part 1 b of the instrument, in this third embodiment, said second portion 1 d also has an alternating arrangement between the cutting edges located on the envelope. The succession of the different zones on the second section 1 d of the instrument in accordance with the third embodiment can be described as follows:

-   -   The first zone of the second portion 1 d immediately adjacent to         the first portion 1 c of the active part 1 b of the instrument         is an off-centre zone 11. For each cross-section 401 b of this         off-centre zone 11, only one of the two diagonally opposed         cutting edges 52, 54 located on the envelope 7 for any         cross-section 40 a of the first portion 1 c—the first cutting         edge 52—is located on the envelope 7 of the instrument, the         second of these cutting edges 54 and the second pair of         diagonally opposed cutting edges 51, 53 being set back within         the envelope 7 (FIG. 10);     -   The second zone is a centred zone 12. For each cross-section 402         b of this centred zone 12, the first and second diagonally         opposed cutting edges 52, 54 are again on the envelope 7 of the         instrument, the second pair of diagonally opposed edges 51, 53         still being set back within the envelope 7 (FIG. 11);     -   The third zone is again an off-centre zone 11. However, in this         zone the second cutting edge 54 diagonally opposed to the first         52 and which was set back within the envelope on the first zone         is now located on the envelope 7, while the first cutting edge         52 is then located within the envelope 7, the second pair of         diagonally opposed cutting edges 51, 53 still being set back         within the envelope 7 (FIG. 12).

The following zone shown in FIG. 13 is similar to the first zone and the alternating arrangement thus extends along the second portion 1 d of the active part 1 b of the instrument.

Thus the instrument in accordance with the third embodiment has two cutting edges 52, 54 which are located on the envelope for any cross-section 40 a of the first portion 1 c of the active part 1 b and for any cross-section 402 b of the centred zones 12 of the second portion 1 d of the active part 1 b but of which at least one of the two is set back within the envelope 7 for any cross-section 401 b of an off-centre zone 11 of the second portion 1 d, two off-centre zones 11 separated by a centred zone 12 not having the same cutting edge on the envelope 7.

In variations, the second pair of diagonally opposed cutting edges 51, 53 could be located on the envelope 7 of the instrument for any cross-section of the active part 1 b or for any cross-section of the first portion 1 c or for any centred cross-section 12 of the second portion 1 d. The second pair of diagonally opposed cutting edges 51, 53 could also follow the same alternating arrangement described above as the first pair of cutting edges 52, 54 along the second portion 1 d of the active part 1 b of the instrument.

The other considerations relating to the first two embodiments remain valid for this third embodiment.

In particular, the first portion 1 c of the active part preferably extends over a length of 3 millimetres from the point 3 of the active part 1 b. In a still more preferred manner, said first portion 1 c has a length of 1 millimetre.

Thus as in the first two embodiments described above, the instrument in accordance with the third embodiment is effective since it has a point 3 which is centred and does not generate beating in the canal and permits precise shaping of said canal. Said instrument is also flexible at the rear owing to the presence of the off-centre zones on the second portion 1 d of its active part 1 b. However, in contrast to the first two embodiments in which the whole of the second portion 1 d is off-centre and thus may generate a beating motion at the rear of the instrument, the alternating arrangement of centred and off-centre zones as described in the third embodiment makes it possible to ensure that the instrument is flexible while avoiding beating by ensuring the shape of the envelope generated by the instrument in rotation. Thus the instrument in accordance with the third embodiment retains all the advantages described in relation to the first two embodiments while reducing the beating motion at the rear of the instrument, a motion which could reduce the precision and speed of treatment of the dental root canal.

The embodiments presented above describe polygonal cross-sections with straight sides. It is clear that said sides could be curved. Consequently, the term “polygonal” should be understood in its general sense meaning “which has a plurality of sides” and covering equally a geometric shape with straight or curved sides.

The instrument in accordance with the invention could obviously have other known features such as variable conicity. Similarly, the flutes can be oriented equally to the right or left or even have a variable pitch.

In a general manner, the active part of an instrument in accordance with the invention has, over is whole length, a polygonal cross-section forming at least two cutting edges. The active part is defined by an essentially conical or cylindrical envelope, the longitudinal axis of which coincides with the axis of rotation of the instrument. The active part comprises a first portion extending from the point of the instrument and which is such that any cross-section of this first portion has its centre of mass on the axis of the envelope and defines at least two cutting edges which are all located on said envelope. The active part also comprises a second portion extending following the first portion towards the rear of the active part and which is such that at least one cross-section of this second portion has a centre of mass which is not located on the axis of the envelope but which is offset with respect to this axis and defines cutting edges, of which at least one is located on the envelope and at least one is located set back within said envelope.

The instrument in accordance with the invention thus has a point which is centred with respect to the axis of rotation of the instrument and a rear part, of which at least one cross-section is off-centre. The centred point makes it possible to follow the initial path of the root canal, to avoid any beating motion within said canal in the vicinity of the point and to ensure dimensioning in the preparation of the apical part of the canal. Moreover, the centred point avoids debris descending towards said apical part of the canal and contributes to successful evacuation thereof. This evacuation is further facilitated by the fact that at least one cross-section of the rear part is off-centre: the debris in fact has more space to be carried out of the canal and the off-centring of at least one cross-section of the rear part of the instrument creates a dynamic effect which lifts the debris out of the canal.

Moreover, an instrument in accordance with the present invention makes it possible to ensure strength in the part close to the point in order thus to reduce the risk of breaking the instrument in its most fragile portion. The flexibility of the instrument is not thereby reduced, since the second portion having at least one off-centre cross-section of the instrument makes the instrument flexible and this flexibility can even be progressive towards the rear of the instrument either by adjusting the off-centring of the second portion of the active part of the instrument as described in the case of the first and second embodiments or by alternating centred cross-section with off-centre cross-section along the second portion of the active part of the instrument as described in the third embodiment. Thus the instrument can be rendered sufficiently flexible to follow the complex contour of a dental root canal.

Finally, the active cutting edges of the point of the instrument, i.e. those which are located on the envelope, ensure a good level of efficacy of the instrument. At the rear, the contact zones of the instrument are reduced since some of the edges are located set back within the envelope of the instrument, which makes it possible to reduce the screwing effect and superfluous forces on the rear part of the instrument.

Thus a high-performance instrument is produced having a good balance between efficacy, flexibility and resistance to breaking and to the screwing effect. 

1. An endodontic instrument comprising: a tapered rod, the rod comprising an active portion, the active portion terminating at a distal tip end of the rod and defining a conical envelope, the conical envelope having a longitudinal axis that coincides with an axis of rotation of the instrument; the active portion comprising a plurality of helical flutes disposed along at least a segment of the active portion, the plurality of helical flutes comprising a first flute, a second flute, a third flute, and a fourth flute; the active portion comprising a plurality of edges, the plurality of edges comprising a first edge intersecting the first flute and the second flute, a second edge intersecting the second flute and the third flute, a third edge intersecting the third flute and the fourth flute, and a fourth edge intersecting the first flute and the fourth flute; the active portion comprising a first portion having a length and a second portion having a length, the second portion being disposed further from the tip end relative to the first portion; the first portion comprising a plurality of first portion cross-sections, the plurality of first portion cross-sections comprising a first cross-section, each of the first edge, the second edge, the third edge, and the fourth edge intersecting the first cross-section, the intersection between the first edge and the first cross-section being located on the envelope; the second portion comprising a plurality of second portion cross-sections, the plurality of second portion cross-sections comprising a second cross-section, wherein a center of mass of the second cross-section is offset relative to the axis of rotation of the instrument, each of the first edge, the second edge, the third edge, and the fourth edge intersecting the second cross-section, the fourth flute being overcut along at least a segment of the second portion such that the third edge and the fourth edge are set back within the envelope along the segment; wherein the plurality of second portion cross-sections further comprises a third cross-section, the third cross-section having a center of mass and being disposed further from the tip end relative to the second cross-section, the distance between the center of mass of the second cross-section and the axis of rotation, considered in proportion to the size of the second cross-section, is less than the distance between the center of mass of the third cross-section and the axis of rotation, considered in proportion to the size of the third cross-section.
 2. The endodontic instrument of claim 1, wherein the length of the second portion extends from the first portion to a proximal end of the active portion, and the fourth flute is overcut along the length of the second portion.
 3. The endodontic instrument of claim 1, wherein the fourth flute is overcut proportionally more, considered relative to the rod's cross-sectional size, over the length of the second portion in a direction defined from the distal tip end toward a proximal end of the active portion.
 4. The endodontic instrument of claim 1, wherein the second cross-section is located at a distal end of the second portion, and the center of mass of the second cross-section is closer, considered in proportion to the size of the respective cross-section, to the axis of rotation than are the centers of mass of every other cross-section along the length of the second portion.
 5. The endodontic instrument of claim 2, wherein the plurality of second portion cross-sections further comprises a third cross-section, the third cross-section having a center of mass and being disposed further from the tip end relative to the second cross-section, the distance between the center of mass of the second cross-section and the axis of rotation, considered in proportion to the size of the second cross-section, is less than the distance between the center of mass of the third cross-section and the axis of rotation, considered in proportion to the size of the third cross-section.
 6. The endodontic instrument of claim 5, wherein the fourth flute is overcut proportionally more, considered relative to the rod's cross-sectional size, over the length of the second portion in a direction defined from the distal tip end toward a proximal end of the active portion.
 7. The endodontic instrument of claim 1, wherein the intersection between the second edge and the first cross-section is located on the envelope, the intersection between the third edge and the first cross-section is located on the envelope, and the intersection between the fourth edge and the first cross-section is located on the envelope.
 8. The endodontic instrument of claim 7, wherein the first cross-section approximates a square, and the second cross-section approximates a rectangle.
 9. The endodontic instrument of claim 1, wherein every cross-section of the first portion has a center of mass located on the axis of rotation of the instrument, and the length of the first portion is between 1 and 3 millimeters.
 10. The endodontic instrument of claim 9, wherein the length of the second portion extends from the first portion to a proximal end of the active portion, and the fourth flute is overcut along the length of the second portion.
 11. The endodontic instrument of claim 10, wherein every cross-section along the length of the second portion has a center of mass that is offset relative to the axis of rotation.
 12. The endodontic instrument of claim 10, wherein the plurality of second portion cross-sections further comprises a third cross-section, the third cross-section having a center of mass and being disposed further from the tip end relative to the second cross-section, the distance between the center of mass of the second cross-section and the axis of rotation, considered in proportion to the size of the second cross-section, is less than the distance between the center of mass of the third cross-section and the axis of rotation, considered in proportion to the size of the third cross-section.
 13. The endodontic instrument of claim 10, wherein the fourth flute is overcut proportionally more, considered relative to the rod's cross-sectional size, over the length of the second portion in a direction defined from the tip end toward the proximal end of the active portion.
 14. The endodontic instrument of claim 9, wherein the plurality of second portion cross-sections further comprises a third cross-section, the third cross-section having a center of mass and being disposed further from the tip end relative to the second cross-section, the distance between the center of mass of the second cross-section and the axis of rotation, considered in proportion to the size of the second cross-section, is less than the distance between the center of mass of the third cross-section and the axis of rotation, considered in proportion to the size of the third cross-section.
 15. The endodontic instrument of claim 14, wherein the second cross-section is located at a distal end of the second portion, and the center of mass of the second cross-section is closer, considered in proportion to the size of the respective cross-section, to the axis of rotation than are the centers of mass of every other cross-section along the length of the second portion.
 16. The endodontic instrument of claim 1, wherein a flexibility of the instrument increases in a direction defined from the tip end toward a proximal end of the active portion.
 17. The endodontic instrument of claim 1, wherein a flexibility of the instrument increases in a direction defined from the tip end toward a proximal end of the active portion.
 18. The endodontic instrument of claim 7, wherein a flexibility of the instrument increases in a direction defined from the tip end toward a proximal end of the active portion.
 19. The endodontic instrument of claim 1, wherein exactly two of the intersections between the second cross-section and the first, second, third, and fourth edges are located on the envelope.
 20. The endodontic instrument of claim 2, wherein exactly two of the intersections between the second cross-section and the first, second, third, and fourth edges are located on the envelope.
 21. The endodontic instrument of claim 1, wherein exactly two of the intersections between the second cross-section and the first, second, third, and fourth edges are located on the envelope.
 22. The endodontic instrument of claim 1, wherein the length of the first portion is between 1 and 3 millimeters, the second cross-section approximates a rectangle, and every cross-section of the first portion approximates a square.
 23. The endodontic instrument of claim 1, wherein at least two of the intersections between the first cross-section and first, second, third, and fourth edges are located on the envelope; and the length of the second portion extends from the first portion to a proximal end of the active portion, and the fourth flute is overcut along the length of the second portion.
 24. The endodontic instrument of claim 23, wherein exactly two of the intersections between the second cross-section and the first, second, third, and fourth edges are located on the envelope.
 25. The endodontic instrument of claim 24, wherein the plurality of second portion cross-sections further comprises a third cross-section, the third cross-section having a center of mass and being disposed further from the tip end relative to the second cross-section, the distance between the center of mass of the second cross-section and the axis of rotation, considered in proportion to the size of the second cross-section, is less than the distance between the center of mass of the third cross-section and the axis of rotation, considered in proportion to the size of the third cross-section.
 26. The endodontic instrument of claim 25, wherein the intersection between the second edge and the first cross-section is located on the envelope, the intersection between the third edge and the first cross-section is located on the envelope, and the intersection between the fourth edge and the first cross-section is located on the envelope.
 27. The endodontic instrument of claim 25, wherein the first cross-section approximates a square, and the second cross-section approximates a rectangle.
 28. The endodontic instrument of claim 25, wherein every cross-section of the first portion has a center of mass located on the axis of rotation of the instrument, and the length of the first portion is between 1 and 3 millimeters.
 29. The endodontic instrument of claim 25, wherein the length of the first portion is between 1 and 3 millimeters, the second cross-section approximates a rectangle, and every cross-section of the first portion approximates a square. 